Two-Parameter SVD: Coalescing Singular Values and Periodicity

We consider matrix valued functions of two parameters in a simply connected region Ω. We propose a new criterion to detect when such functions have coalescing singular values. For generic coalescings, the singular values come together in a “double cone”-like intersection. We relate the existence of any such singularity to the periodic structure of the orthogonal factors in the singular value decomposition of the one-parameter matrix function obtained restricting to closed loops in Ω. Our theoretical result is very amenable to approximate numerically the location of the singularities.